8 COMPLETE HUBERT ALGEBRAS 365
we take any sequence of pairwise orthogonal, irreducible self-adjoint idempo-
tents belonging to b and such that Xi = ]T xlelt t (15.8.10). Now suppose that
i
the <?m> i have been defined for m g n in such a way that they are pairwise
orthogonal and belong to b, and are such that the xm with m <^ n belong to the
closure an c b of the left ideal which is the sum of the ideals Aemt i for all
m g n and / 6 Im for each m. Let x'n+l be the orthogonal projection of xn+1
on ax n b; then take for (en + it i)fein+1 a sequence of pairwise orthogonal
irreducible self-adjoint idempotents which belong to a1 n b and are such
that xrn+1 = ]T x'n+1en+lt t (15.8.10). It is clear that the double family (ent t),
i
arranged in a single sequence, has the required properties, by virtue of
(15.8.9) and (6.4.2).
This theorem applies in particular when b = A. In this case we get a
decomposition of A as a Hilbert sum of minimal left ideals. It should be
remarked that in general there will exist infinitely many such decompositions
(see later (15.8.14)). More precisely:
(15.8.11.1) Suppose that A is separable, and let I be a minimal left ideal of A.
Then there exists a decomposition of A as a Hilbert sum of minimal left ideals
ln, such that l^ - I.
Apply (15.8.11) to b = I1.
(15.8.12) Let e, e' be two irreducible self-adjoint idempotents, and let I = Ae,
V = Ae' be the corresponding minimal left ideals.
(i) Every homomorphism of the A-module I into the A-module I' is of the
formfa: xt-*xa, where a e eAe' = eA n Ae'\ it is either zero or bijective, and
the mapping a\-*fa is an isomorphism of the complex vector space eAer onto
HomA(I, F) such thatfab =fb °fa.
(ii) The C-algebra eAe, isomorphic to EndA(I), is afield, equal to Ce (and
therefore isomorphic to C).
(iii) If I and V are not isomorphic as A-modules, then e and e' (and
consequently I and V) are orthogonal, and IV = I'l = {0}. If I and V are iso-
morphic as A-modules, then eAe' is a complex vector space of dimension 1,
and ir = I'.
(iv) IfxeA, then Ix is a left ideal which is either zero or isomorphic (as
A-module) to I.
(i) If g : I -ğI' is an A-module homomorphism and a = g(e), then for
all xel we have g(x) = g(xe) = xg(e) = xa (15.8.3(iii)). Since a el' we^ xen, and for all x, y in b we have (x\y) = £ (xen \ yen).